Abstract

In this paper, we consider some combinatorial and statistical aspects of the popular “Powerball” lottery game. It is not
difficult for students in an introductory statistics course to compute the probabilities of winning various prizes,
including the “jackpot” in the Powerball game. Assuming a unique jackpot winner, it is not difficult to find the expected
value and the variance of the probability distribution for the dollar prize amount. In certain circumstances, the expected
value is positive, which might suggest that it would be desirable to buy Powerball tickets. However, due to the extremely
high coefficient of variation in this problem, we use the law of large numbers to show that we would need to buy an
untenable number of tickets to be reasonably confident of making a profit. We also consider the impact of sharing the
jackpot with other winners.

1. Introduction

Our motivation for writing this paper was the popular Powerball Lottery game, which is played in 23 states including our
home state of Kentucky. For $1, a player chooses five different numbers between 1 and 53 (inclusive). In addition, the
player chooses a powerball number between 1 and 42 (inclusive). The player wins the jackpot if his or her
five numbers match the five numbers between 1 and 53 chosen without replacement in a random drawing and if the
player’s powerball number also matches the powerball number between 1 and 42, also drawn randomly. The jackpot is a
varying large amount of money which will add its value from one drawing to the next if there is no winner. There are also
other ways to win smaller prizes, as will be indicated later. The jackpot prize is typically several million dollars.
According to the Multi-State Lottery Association (2003), Andrew Whittaker of
West Virginia won the largest undivided lottery prize in U.S. history on Christmas Day, 2002. He won a jackpot of $314.9
million; by choosing the “cash” option, he actually won $170.5 million before taxes.

There exist a large number of books on lotteries. The vast majority of these books, such as
Howard (1997), are so full of inane mathematical and statistical errors as to
be not worth the paper they are printed on. A rare exception is Henze and Riedwyl (1998).
This book is one of the few mathematically accurate and honest books that exists on lotteries.

An r/s lottery consists of choosing r distinct numbers from S = {1, 2, ..., s}. To win the jackpot,
one must match all r numbers drawn without replacement from S. The probability of winning the jackpot is
. Many lotteries in various U.S. states and European countries are
r/s lotteries; it is very common for r = 6 and s to be an integer between 40 and 60. For example, in
the state of Kentucky, the Lotto South game is a 6/49 lottery. The number of possible tickets is
= 13,983,816.

A phenomenon of lottery games is that the number of tickets purchased increases greatly as the size of the jackpot increases.
If several drawings go by without a jackpot winner, the jackpot goes to many millions of dollars, which encourages regular
lottery players to buy more tickets than usual and infrequent players to participate. This is a desirable situation for
lottery commissions, who keep a large percentage of the money raised in ticket sales to fund various governmental programs.

However, with millions of players per drawing, most r/s lotteries do not go very long without a jackpot winner. In
order to build up a larger jackpot, which encourages more players, there needs to be even more possible combinations,
making the game harder to win. Now consider a r1 / s1 +
r2 / s2 lottery,
which involves choosing r1 distinct numbers from S1 =
{1, 2, ..., s1} and r2 distinct numbers from
S2 = {1, 2, ..., s2}. Typically,
r2 = 1 and that number is referred to as a supernumber or a powerball. The popular
Powerball lottery game is a 5/53 + 1/42 lottery. Mega Millions is a similar game played in 11 states; it is a 5/52 + 1/52
lottery. The calculations throughout this paper will focus on Powerball, but could be replicated for Mega Millions.

2. Powerball Odds

In addition to the multi-million dollar prize won by matching all 5 regular numbers and the powerball, there are several
other scenarios where one can win a lesser prize. For instance, matching the first five numbers but failing to match the
powerball still wins a prize of $100,000. There are a total of 9 winning scenarios; these scenarios and their associated
winnings and probabilities are given in the table below, where n is the number of the first five numbers that are
matched. Also in the table is the 10th and most common scenario, in which the $1 is lost.

Table 1. Probabilities and Prizes of Powerball Scenarios.

Scenario

n

Powerball?

Prize

Probability

#1

5

Y

Jackpot

1:120,526,770

#2

5

N

$100,000

41:120,526,770

#3

4

Y

$5,000

240:120,526,770

#4

4

N

$100

9,840:120,526,770

#5

3

Y

$100

11,280:120,526,770

#6

3

N

$7

462,480:120,526,770

#7

2

Y

$7

172,960:120,526,770

#8

1

Y

$4

972,900:120,526,770

#9

0

Y

$3

1,712,304:120,526,770

#10

0, 1, 2

N

None

117,184,724:120,526,770

We will use combinatorial reasoning to verify the probabilities above; this is an exercise accessible to a student in a
first course in statistics and/or probability. It is easy enough to count the total number of possibilities for the random
drawing. There are five numbers chosen without replacement from the set S1 =
{1, 2, ..., 52, 53} and one number chosen from the set S2 = {1, 2, ..., 41, 42}. This gives us

possibilities.

Now consider the number of possible ways a player could get exactly n of the first numbers right (where 0
n 5) and also get
the powerball correct. There are five correct numbers, of which the player gets n, and there are 48 incorrect
numbers, of which the player gets 5 - n. There is only one way to get the powerball number right. This gives

possibilities.

Finally, consider the number of possible ways a player could get exactly n of the first five numbers right (where
0 n 5) but get the
powerball number wrong. There are five correct numbers, of which the player gets n, and there are 48 incorrect
numbers, of which the player gets 5 - n. There are 41 ways to get the powerball number wrong. This gives

possibilities.

3. Expected Value and Variance

It is not difficult to find the expected value and the variance of a discrete probability distribution. If X is a
discrete random variable and f(x) is the value of its probability distribution at x, then the expected
value is

the variance is

and the standard deviation is .

Suppose that X represents the number of dollars we win (or lose) when we purchase one Powerball ticket. If our
ticket wins no prize, then X = -1 (this is scenario #10); if we are in scenario #9, then X = 3 - 1 = 2 and so
on through scenario #2. One difficulty is that the value of X when scenario #1 (i.e. winning the jackpot) occurs is
not known. Let j represent the value of the jackpot. For now, we will ignore the possibility that there could be
i > 1 winners of the jackpot, where our winnings would be $j/i. We are interested in finding when E(X)
> 0, since a player might have the notion that it is beneficial to buy lottery tickets when the expectation is positive.
We will show later that, from a probabilistic viewpoint, this notion is naive.

Let us compute E(X) and Var(X), ignoring realities like federal and state taxes and the option of taking the
jackpot in one cash payment, rather than an annuity paid over several years. Let n be the number of the first five
numbers matched, where 0 n 5. Referring to Table 1, we summed
the probabilities for the winning scenarios and determined that the chance of winning some prize was 3,342,046 out of
120,526,770. This is only about 2.8%. Therefore, the other 117,184,724 ticket combinations (about 97.2%) will result in
the loss of one dollar.

The expected value is

For E(X) > 0, the jackpot needs to be j > $99,638,177. If the jackpot is exactly j = $99,638,177 and
there is i = 1 winner, the standard deviaton is = 9075.97.

So, at first glance, we may think that it is beneficial to play Powerball whenever j $99,638,177. Unfortunately, the following issues all work to reduce the
value of the jackpot:

Federal and state taxes

The fact that the jackpot winning is much less than the advertised value if one chooses to take the prize in a single
lump-sum cash payment rather than a multi-year annuity

The chance of multiple jackpot winners

Let us also assume that the winner will choose the one time cash payment if he/she hits the jackpot. This will reduce the
prize considerably from the advertised jackpot value. As an example, recall the case of Andrew Whittaker of West Virginia.
According to the Multi-State Lottery Association (2003), he was the sole winner
of the $314.9 million Powerball drawing on 12/25/2002. Mr. Whittaker chose the lump-sum payout, which was $170.5 million.
After taxes, he was left with $113.4 million. The lump-sum payout is 170.5/314.9 or about 54% of the advertised jackpot.
After taxes, Whittaker kept 113.4/170.5 or about 66.5% of the lump-sum payout. So Whittaker only cleared about 113.4/314.9
or about 36% of the advertised jackpot.

Let us assume that these percentages are typical for a large Powerball jackpot. If the advertised jackpot is j
dollars, a single jackpot winner will only win 0.36j dollars. Taxes will also need to be paid if one wins a Scenario
#2 or #3 prize. We will assume that we will keep 70% of our winnings after taxes in those situations. In Scenarios #4-9,
let us assume that the small prizes are countered by a sufficient number of losing tickets such that the player has a
yearly loss and thus does not need to pay taxes on the winnings. The implication is that the value of j such that
E(X) > 0 will increase.

For E(X) > 0, the jackpot needs to be j > $281,189,146.40. If the jackpot is exactly j =
$281,189,146.40 and there is i = 1 winner, the standard deviaton is
= $9,220.69.

If the advertised jackpot is j = $315 million (about what it was for Whittaker’s win) and we assume only i = 1
jackpot winner, then the expected value per dollar ticket is

with standard deviation = $10,329.39.

So it appears that it might be advisable to purchase lottery tickets in the relatively rare situations when the
advertised jackpot exceeds approximately $281.2 million due to the positive expectation. However, we will show that this is
not the case.

Let us compare large jackpot Powerball to the common and simple casino game Roulette. In American casinos, the Roulette
wheel has 38 slots: 36 of which are numbered 1, 2, ..., 36 and are colored either red or black, and 2 of which are numbered
0, 00 and are colored green. A common bet is to select one integer from W = 1, 2, ..., 36. If your choice comes up
on the next spin of the wheel, then you are paid out at odds of 35:1; otherwise, you lose.

It is simple to compute the expected value and standard deviation of this game. Let Y be the amount won or lost on a
$1 bet.

Table 3. Probability Distribution for a Bet on a Single Number in Roulette

Result of Spin

Y

f(y)

Win

$35

1/38

Lose

-$1

37/38

The expected value is

per spin, with = $5.763. The expectation is only negative from the
players’ standpoint; it is positive $0.053 per spin from the casino’s perspective. Notice that this expectation is
approximately one-half the expected value of Powerball with a $315 million jackpot and that the standard deviation is much
smaller (by several orders of magnitude) in Roulette than Powerball.

4. The Law of (very) Large Numbers

Most students realize that games of chance such as Roulette are profitable (even after expenses of doing business) for a
casino in the long-run; the best a player can hope for is to be “lucky” in the short-term. Shouldn’t the same hold true
for playing a game of chance such as Powerball if we are disciplined enough to only play when the expected value is
positive? After all, the weak law of large numbers tells us that if a random sample is drawn from a probability
distribution, the probability that the sample mean will deviate from
by less than some arbitrary quantity approaches 1 for a sufficiently large sample size n.

Let f(x) be a probability density function with mean and finite
variance . Let be
the sample mean of a random sample of size n from f(x). Choose constants and such that
> 0 and 0 < < 1.
If n is an integer where

(1)

then

(2)

Typically the constant is chosen to be very close to zero and the
theorem is used to demonstrate that as , the sample mean will eventually
get - close to . For
example, suppose we have a distribution with mean = 0.5 and
= 0.25. An example of a distribution with these values for the mean
and the variance is the Bernoulli distribution. It mathematically models the flipping of a fair coin, letting X = 1
when we obtain heads and X = 0 we we obtain tails.

We will flip a fair coin n times, count the number of heads obtained, and divide by n. Suppose we want to be
at least 95% sure that the sample mean will be between .499 and
.501. We have = 0.25,
= 1 - 0.95 = 0.05, and = 0.01; therefore

To be only at least 50% sure, = 1 - 0.5 = 0.5 and the required sample
size would be n 5000.

Figure 1 shows the weak law of large numbers in action for a simulation of 50000
coin tosses implemented with the R statistical computing package.

While it is customary to take the constant to be very close to zero,
we will instead take to be equal to the mean of our probability
distribution; that is, =
Substituting for in
equation(2) yields

This inequality tells us that if n is sufficiently large as defined in inequality (1),
then there is at least a 100(1 - )% chance that the sample mean
will be between 0 and 2.
For large n, this essentially is the probability that > 0, since
the probability that > 2
is virtually zero for large n. In the context of a game of chance, this is the probability that our winnings are
greater than our losses after n trials of the game.

To determine how large n needs for there to be a probability of at least 1 -
of having positive winnings, substitute =
into inequality (1):

(3)

In statistics, we define the coefficient of variation, or CV, as the ratio of the standard deviation to the
mean of a distribution. That is,

Thus, we see that the sample size n required to be confident of positive winnings is proportional to the square
of the coefficient of variation.

Now reconsider random variables X and Y, which correspond to Powerball and Roulette, respectively. We will
compute the coefficients of variation for both X and Y. The computation for Powerball will assume a jackpot
of $315 million (i.e. Whittaker’s Christmas Day win) and a single jackpot winner.

Notice that the coefficent of variation for Powerball (even in a very large jackpot) is larger than the CV for
Roulette by about 3 orders of magnitude. This means, of course, that the squared coefficients of variation for the two
games will differ by about 6 orders of magnitude. To put it another way, the sample size required to be confident of
positive winnings as a Powerball player will be approximately 1,000,000 larger than that of a casino offering Roulette. To
be more precise, if we set = 0.05 to be at least 95% confident that
is positive (for the casino), we need the following sample sizes:

Even if we relax the level of confidence for having 0 in Powerball, huge samples are still required. For 50% confidence, we
need

and for
10% confidence, we need

While nY is fairly large, it is not unreasonable that over time, there will be hundreds of
thousands and possibly even millions of bets placed at a roulette wheel. Even taking into account the casino’s expenses,
over the long run games such as Roulette will be profitable.

However, nX is approximately 200 billion, which is ludicrously large. This indicates that to
be at least 95% confident of winning money on the Powerball, even if we are disciplined enough to only play when the
expected value is positive and fortunate enough to be the unique winner when we hit the jackpot, we will have to play
hundreds of billions of times. This would require hundreds of billions of dollars (which we don’t have) and hundreds of
billions of opportunities to play Powerball when the jackpot is high. Even if we buy hundreds or thousands or even millions
of tickets when the expectation is positive, we will probably die long before we are ahead. If we lowered the confidence
level to 50% or even 10%, we would still expect to need to play 10 to 20 billion times to realize a profit.

Even if we did have hundreds of billions of dollars at our disposal to play Powerball, we probably wouldn’t want to.
Risking $200 billion for the chance to win a prize of even $200 million would be proportional to a person with a yearly
income of $50,000 risking that entire salary on a game of chance for the opportunity to win a $50 prize.

To give a graphical sense of the typical fortunes one would have either playing Powerball or running a Roulette wheel, we
have simulated each game 50000 times. We have assumed each Powerball ticket or Roulette bet is $1 and that we are playing
Powerball with a $315 million jackpot with a unique winner. Figure 2 and
Figure 3 show the results of the simulations for Powerball and roulette,
respectively.

The unfortunate Powerball player, after 50000 plays, had lost $42,892, or about 85.8 cents per $1 ticket. Notice the graph
closely resembles a straight line with slope -1 and y - intercept 0. The line has occasional “jumps” (virtually
imperceptible to the naked eye) where the player won one of the minor prizes.

In contrast, notice there are many more ups and downs with Roulette. In this simulation, the casino ended up $4496 ahead
after 50000 spins, an average of about 8.9 cents per spin, which is somewhat higher than the theoretical mean of
= 5.3 cents per spin. Over thousands of more plays, the sample mean would
fall back to . There are periods over thousands of plays where the casino
(or you) has both lost money and won money. In the end, the weak law of large numbers guarantees the casino will make money
(and the player will lose money) at a rate of per dollar bet. But the much
smaller coefficient of variation gives the player a reasonable chance of being ahead in the short-term.

We certainly would encourage instructors and students to design their own simulations. One could choose to reproduce our
simulations of Powerball and roulette, or choose to reproduce different lotteries or different casino games. It would be
interesting to compute the mean, variance, and coefficient of variation for other games of chance to compare with our values
for Powerball and roulette.

It is doubtful that anyone would play casino games such as roulette, blackjack, slot machines, craps, etc. if the
coefficient of variation was of order of magnitude 6. The average player would never experience a win and would eventually
quit playing, since the prizes available in this game are not monumentally large. However, people seem perfectly happy to
risk a dollar (or many dollars) per week on the lottery although the
CV is monstrously large; the faint glimmer of hope of becoming an instant multi-millionaire is enough to keep
millions of players interested.

5. Multiple Jackpot Winners

So far in our discussion, we have made the simplifying assumption that there is a unique jackpot winner. This is not a
terrible assumption, since the empirical evidence gathered from 597 Powerball drawings from 11/5/1997 until 7/26/2003
yielded the following results:

Table 4. Number of Jackpot Winners, Powerball 1997-2003.

Number ofJackpot Winners

Number ofOccurences

0

538

1

49

2

9

3

0

4

1

5+

0

So about 90% of Powerball drawings fail to yield a jackpot winner, which leads to a larger jackpot available in the next
drawing. This serves the interests of lottery commissions well. Since the Powerball game is very difficult to win, there
is often an opportunity to build a large jackpot and stimulate a frenzy of ticket buying. Earlier, we assumed that all
jackpot winners were unique to simplify the computation of the expected value and variance of Powerball. Empirically, we
see that multiple jackpot winners is a relatively uncommon event. Only 10 of 597 drawings (about 1.7%) and 10 of 59
jackpot wins (about 17%) featured multiple winners.

We have already seen that the necessary jackpot for a positive expectation is rather large, about $281 million, for
Powerball. This fact, coupled with the large variance and huge coefficient of variation, led us to conclude that one would
need to play Powerball hundreds of billions of times to reasonably expect to make a profit. Practically, this is not
possible.

A more advanced problem is to determine the probability of having exactly i jackpot winners from among k
players who each randomly choose one of n possible ticket combinations. For Powerball, n =
120,526,770. Equivalently, we can think of this as a problem where k players each randomly choose one number (with
repetition allowed) from the set S = {1, 2, ..., n - 1, n}.

We will name the players in this game P1, P2, ...,
Pk - 1, Pk, where you are player
Pk. What is the probability that there will be exactly i players including
you who pick the winning number? First, since we are assuming you are a winner, we need to decide how many ways
there are to have i - 1 winners out of the first k - 1 players. This number is
. Next, consider how many choices are possible for the remaining k -
i players who do not win. Each such player can choose from among n - 1 numbers (i.e. all numbers except the winning
number), which gives a total of (n - 1)k - i possibilities. Finally, there are a grand total of
nk choices that the k players in this game can make. Thus the probability that there are exactly
i winners including you is:

Now say the payoff is $j split evenly among the winners. Then your expected winnings in this game would be (we are
ignoring, for now, taxes and other factors that serve to reduce j for now):

(4)

If k and n are large, as they will be for Powerball, then numbers like nk,
(n - 1)k - i, and could be difficult to
compute. In what follows, we suggest one approach for analyzing this expectation (students and instructors are encouraged
to consider other approaches). After a bit of
algebraic manipulation, (4) can be written as:

(5)

Now we can use the binomial theorem to obtain a simplified exact expression of (5).

Binomial Theorem For positive integer k,

(6)

We will let a = 1 - 1/n, b = 1/n, and subtract off the 0th term, obtaining

Now let us introduce our Powerball particulars into the computations. The expected value of X, the average win/loss
per Powerball ticket purchase, will be approximately equal to the expected winnings when we hit the jackpot, which we will
call W, plus the expected winnings of the smaller prizes and no prize given in Scenarios #2 - #10, which we will
call V. That is, E(X) = E(V) + E(W), where

and

We use 0.36j instead of j since we expect to only take home 36% of the advertised jackpot value when we take
the lump-sum payment and factor in taxes.

So in order to have a positive expected value, we need to solve the following inequality for j:

(7)

In inequality (7), we know n = 120,526,770. The table below finds
the minimum necessary jackpot j for positive expected value for various values of k, assuming the possibility
of multiple winners.

Recall that the minimum jackpot necessary for a positive expectation, assuming a unique winner, was

j = $281,189,146.40

It is typical for the Powerball lottery to have about 10 to 20 million players for the smaller jackpots (i.e. drawings held
after wins) but at least 50 million players when the jackpot has been built up after several successive drawings without a
jackpot winner. When we factor in the possibility of multiple jackpot winners, the advertised jackpot value needs to be
very large (usually over $300 million!) to have a positive expectation on the purchase of a ticket. Coupling this with the
application of the law of (very) large numbers from the previous section is quite sobering. It is just not rational to
play Powerball with any expectations of winning money, even if we limit our play to large jackpot situations.

6. Factors in Ticket Purchase

Our calculations in the previous sections implicitly made a very large assumption which is false. We were assuming that we
select a ticket combination at random, which we could very well do by selecting “Quick-Pick” or using our own random
number generator. But we assumed that all of our fellow Powerball players are picking their combinations at random, which
is not the case. Henze and Riedwyl (1998) state that quick-pick sales
make up approximately 62% of ticket sales, and about that same percentage of jackpot winners used quick-pick. This means
that about 38% of tickets purchased involved the person manually choosing their numbers. According to the FAQ at
www.powerball.com, the figures are about 70%/30% in favor of quick-pick for
powerball. Again, about 70% of jackpot winner in Powerball (41 of the first 59) used quick-pick.

Kadell and Ylvisaker (1991) were granted access from a lottery commission to
data showing the number of purchases of each combination. They noted that certain combinations are purchased much more
often that would be expected by chance. In fact, some combinations have been observed to have been purchased hundreds or
even thousands of times in a single lottery. For example, the most popular combination in the October 29, 1988 drawing of
the California Lotto 6/49 was 7-14-21-28-35-42, which was purchased 16,771 times. We could speculate the incredible
popularity of this combination was due to the fact that we start with the “lucky” number 7, and then choose the first 6
multiples of 7.

Chapter 5 of Henze and Riedwyl (1998) discussed many popular strategies of
players, which are therefore foolish since buying popular combinations increases the likelihood of splitting a jackpot with
several others. Popular strategies include (not a comprehensive list):

Choosing arithmetic progressions (e.g. 1-2-3-4-5-6 or 2-5-8-11-14-17)

Choosing winning combinations from previous draws

Modifying previous winning combinations (e.g. adding 1 to each number in a previous winning combination)

Choosing “hot” or “cold” numbers (a statistically nonsensical strategy suggested in many of the lay books about
lotteries)

Choosing powers of 2 (e.g. 1-2-4-8-16-32)

Choosing perfect squares (e.g. 1-4-9-16-25-36)

Choosing all prime numbers (e.g. 2-3-5-7-11-13)

Choosing Fibonacci numbers (e.g. 1-2-3-5-8-13)

Choosing only numbers that are less than or equal to 31; many people choose numbers based on birthdays, anniversaries,
etc.

In general, using any simple rule to choose your numbers is foolish, since it is likely that others will use the
same rule. Instead, you want the combination(s) that you purchase to be purchased only by you, in order to avoid
splitting the jackpot. Henze and Riedwyl (1998) suggest that “quick-pick” is
a simple way to make it more probable that you will avoid buying one of the popular combinations and also discuss some
more sophisticated ideas for selecting a combination that is likely to be “unpopular”. However, their discussion is
mostly focused on 6/s lotteries without a supernumber. Since Powerball has a supernumber and the chance of
hitting the jackpot is so miniscule, it hardly seems worth the trouble to go beyond “quick-pick” in your quest for a
unique combination.

The good news is that if we avoid choosing some of the particularly popular combinations (as listed above), we should be
able to somewhat reduce the probability of sharing a jackpot. Between 11/5/1997 and 7/26/2003, the largest number of
distinct winners in Powerball was 4, which occurred in the drawing of August 15, 2001. It would be very interesting to a
neutral spectator to observe a lottery with dozens or even hundreds of jackpot winners if one of the very popular
combinations were to occur. Of course, the winners would be annoyed that their lucky moment was not as rewarding as it
could have been!

7. Conclusion

We have investigated various combinatorial and statistical aspects of the popular lottery game known as Powerball. These
aspects can be used as motivating examples in statistics and probability courses at varying levels of mathematical
sophistication. The simple counting arguments made in section 2 are used to develop
a reasonable probability distribution or model describing one’s chances in Powerball. These results are accessible to even
a non-calculus based introductory course in statistics. These students are also equipped to follow through the results of
section 3, where the expected value and variance for the probability distribution
are found. Even if one stops at this point, these students can see that the jackpot needs to be very large before there is
any hope of Powerball having a positive expectation from the player’s standpoint. This would also be a great opportunity
to introduce the underrated coefficient of variation, which is used in section 4.

The subsequent sections are mathematically more sophisticated and one may wish to avoid or at least “handwave” these
results in the non-calculus course. However, students enrolled in a standard post-calculus course in probability and
statistics, such as an engineering statistics or a typical undergraduate sequence in mathematical statistics should be able
to follow section 4, where it is shown that the law of large numbers often
requires “large” to be very large. One who follows the lead of
Rossman, Chance, and Ballman (2000) and uses an activity-oriented approach
might have the students re-do my simulations of Roulette, Powerball, or other games of chance for themselves.
Section 5 removes the assumption that there is a unique jackpot winner.

Section 6 is a non-technical section that is accessible to any level of student.
The instructor might wish to point out that all possible tickets are equally likely to be chosen in the drawing, despite
the poor advice of self-proclaimed lottery “experts” like Howard (1997).
However, as is thoroughly discussed by Henze and Riedwyl (1998), it is desirable
to avoid certain combinations. The instructor might also wish to discuss why using randomization via the
“Quick-Pick” option is an easy way for the player to avoid choosing a popular combination. The class could discuss why
Henze and Riedwyl are correct in advocating “Quick-Pick” for those who must play and why Howard is wrong in her
condemnation of “Quick-Pick” and advocacy of finding “hot” and “cold” numbers.

Acknowledgements

The authors would like to thank the editor and the referees for their comments on an earlier version of this paper that led
to an improved final version. In particular, the mathematics in Section 5 is now
much simpler and more elegant due to a referee.